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G = C22.D36order 288 = 25·32

1st non-split extension by C22 of D36 acting via D36/D18=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22.2D36, C23.6D18, (C2×Dic9)⋊C4, (C22×D9)⋊C4, C91(C23⋊C4), C22⋊C41D9, (C2×C6).2D12, C6.9(D6⋊C4), (C2×C18).28D4, C22.3(C4×D9), C2.4(D18⋊C4), (C22×C6).35D6, C18.D41C2, C18.2(C22⋊C4), C22.8(C9⋊D4), C3.(C23.6D6), (C22×C18).5C22, (C2×C6).2(C4×S3), (C9×C22⋊C4)⋊1C2, (C2×C18).1(C2×C4), (C2×C9⋊D4).1C2, (C3×C22⋊C4).1S3, (C2×C6).66(C3⋊D4), SmallGroup(288,13)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C22.D36
C1C3C9C18C2×C18C22×C18C2×C9⋊D4 — C22.D36
C9C18C2×C18 — C22.D36
C1C2C23C22⋊C4

Generators and relations for C22.D36
 G = < a,b,c,d | a2=b2=c36=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >

Subgroups: 420 in 78 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22 [×3], C22 [×3], S3, C6, C6 [×3], C2×C4 [×3], D4 [×2], C23, C23, C9, Dic3 [×2], C12, D6 [×2], C2×C6 [×3], C2×C6, C22⋊C4, C22⋊C4, C2×D4, D9, C18, C18 [×3], C2×Dic3 [×2], C3⋊D4 [×2], C2×C12, C22×S3, C22×C6, C23⋊C4, Dic9 [×2], C36, D18 [×2], C2×C18 [×3], C2×C18, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C2×Dic9, C2×Dic9, C9⋊D4 [×2], C2×C36, C22×D9, C22×C18, C23.6D6, C18.D4, C9×C22⋊C4, C2×C9⋊D4, C22.D36
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, C23⋊C4, D18, D6⋊C4, C4×D9, D36, C9⋊D4, C23.6D6, D18⋊C4, C22.D36

Smallest permutation representation of C22.D36
On 72 points
Generators in S72
(1 54)(3 56)(5 58)(7 60)(9 62)(11 64)(13 66)(15 68)(17 70)(19 72)(21 38)(23 40)(25 42)(27 44)(29 46)(31 48)(33 50)(35 52)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 44 54 27)(2 43)(3 25 56 42)(4 24)(5 40 58 23)(6 39)(7 21 60 38)(8 20)(9 72 62 19)(10 71)(11 17 64 70)(12 16)(13 68 66 15)(14 67)(18 63)(22 59)(26 55)(28 36)(29 52 46 35)(30 51)(31 33 48 50)(34 47)(37 61)(41 57)(45 53)(65 69)

G:=sub<Sym(72)| (1,54)(3,56)(5,58)(7,60)(9,62)(11,64)(13,66)(15,68)(17,70)(19,72)(21,38)(23,40)(25,42)(27,44)(29,46)(31,48)(33,50)(35,52), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,44,54,27)(2,43)(3,25,56,42)(4,24)(5,40,58,23)(6,39)(7,21,60,38)(8,20)(9,72,62,19)(10,71)(11,17,64,70)(12,16)(13,68,66,15)(14,67)(18,63)(22,59)(26,55)(28,36)(29,52,46,35)(30,51)(31,33,48,50)(34,47)(37,61)(41,57)(45,53)(65,69)>;

G:=Group( (1,54)(3,56)(5,58)(7,60)(9,62)(11,64)(13,66)(15,68)(17,70)(19,72)(21,38)(23,40)(25,42)(27,44)(29,46)(31,48)(33,50)(35,52), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,44,54,27)(2,43)(3,25,56,42)(4,24)(5,40,58,23)(6,39)(7,21,60,38)(8,20)(9,72,62,19)(10,71)(11,17,64,70)(12,16)(13,68,66,15)(14,67)(18,63)(22,59)(26,55)(28,36)(29,52,46,35)(30,51)(31,33,48,50)(34,47)(37,61)(41,57)(45,53)(65,69) );

G=PermutationGroup([(1,54),(3,56),(5,58),(7,60),(9,62),(11,64),(13,66),(15,68),(17,70),(19,72),(21,38),(23,40),(25,42),(27,44),(29,46),(31,48),(33,50),(35,52)], [(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,53)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,44,54,27),(2,43),(3,25,56,42),(4,24),(5,40,58,23),(6,39),(7,21,60,38),(8,20),(9,72,62,19),(10,71),(11,17,64,70),(12,16),(13,68,66,15),(14,67),(18,63),(22,59),(26,55),(28,36),(29,52,46,35),(30,51),(31,33,48,50),(34,47),(37,61),(41,57),(45,53),(65,69)])

51 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E6A6B6C6D6E9A9B9C12A12B12C12D18A···18I18J···18O36A···36L
order122222344444666669991212121218···1818···1836···36
size11222362443636362224422244442···24···44···4

51 irreducible representations

dim11111122222222222444
type++++++++++++
imageC1C2C2C2C4C4S3D4D6D9C4×S3D12C3⋊D4D18C4×D9D36C9⋊D4C23⋊C4C23.6D6C22.D36
kernelC22.D36C18.D4C9×C22⋊C4C2×C9⋊D4C2×Dic9C22×D9C3×C22⋊C4C2×C18C22×C6C22⋊C4C2×C6C2×C6C2×C6C23C22C22C22C9C3C1
# reps11112212132223666126

Matrix representation of C22.D36 in GL4(𝔽37) generated by

36000
03600
0010
0001
,
36000
03600
00360
00036
,
002031
00626
22600
111300
,
242600
21300
00266
001711
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[0,0,2,11,0,0,26,13,20,6,0,0,31,26,0,0],[24,2,0,0,26,13,0,0,0,0,26,17,0,0,6,11] >;

C22.D36 in GAP, Magma, Sage, TeX

C_2^2.D_{36}
% in TeX

G:=Group("C2^2.D36");
// GroupNames label

G:=SmallGroup(288,13);
// by ID

G=gap.SmallGroup(288,13);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,346,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^36=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations

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